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Resonance impurity radiation in CuGaS2 crystals
Optical Materials 30 (2007) 451–456 www.elsevier.com/locate/optmat
Resonance impurity radiation in CuGaS2 crystals N.N. Syrbu a, L.L. Nemerenco b
a,*
, V.E. Tezlevan
b
a Technical University of Moldova, MD-2004 Chisinau, Republic of Moldova Institute of Applied Physics, Academy of Sciences of Moldova, MD-2028, Chisinau, Republic of Moldova
Received 3 October 2006; received in revised form 6 December 2006; accepted 7 December 2006 Available online 26 January 2007
Abstract In this work the reflectivity spectra of the CuGaS2 crystals have been investigated in the EIIc polarization and the absorption spectra were studied in the E ? c polarization at the 10 K. The parameters of the excitons and the band gaps have been determined. The low ˚ and 4965 A ˚ temperature (10 K) luminescence spectra of the CuGaS2 crystals have been investigated under the excitation by 4880 A Ar+ laser lines. A model of optic transitions between the levels of excitons bound on neutral donor is proposed. 2007 Elsevier B.V. All rights reserved. Keywords: CuGaS2; Photoluminescence; Absorption; Reflection; Exciton bound on neutral donor (acceptor)
1. Introduction In a number of crystals the resonance luminescence is studied, where laser light is elastically scattered in the crystal leaving donors or acceptors in the excited state [1–8]. The investigation of radiation of bound excitons under the excitation of laser light with the energy close to or equal to the bound exciton (resonance excitation) presents especial interest. The CuGaS2 compound is characterized by anisotropy of optic properties and intensive luminescence on bound excitons. This compound belongs to the chalcopyrite group with space group D12 2d [9–12]. In these crystals resonance Raman scattering of exciton polaritons [13–17], interference of additional exciton waves, spatial dispersion of exciton polaritons [18–20] and biexciton radiation [21] are found. In CuGaS2 crystals three exciton series A, B, C are found [12,17–19]. Interaction of electrons of the conduction band C6 and holes of band C7 is determined by the product of irreducible representations C1 · C6 · C7 = C3 + C4 + C5. As a result of this interaction in the longwave region the C4 exciton allowed in EIIc polarization, the C5 exciton allowed in E ? c polarization and the C3 *
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exciton forbidden in both polarizations are formed. The polariton radiation of the C4 exciton was found and the Raman scattering under the excitation with various laser lines which energy exceeds the longitudinal exciton energy was investigated [9–12]. A series of narrow lines attributed to bound excitons was found in the long-wave region from the transverse exciton frequency [13–16]. However, the bound excitons have not been studied in detail and the structure of zero phonon lines and their replicas have not been revealed. The luminescence excited by the energy less than the band gap (resonance luminescence of bound excitons) has not been reported. We have studied the luminescence of CuGaS2 crystals at excitation energy E < Eg and E being less than the energy of free exciton ground state radiation. We have considered the case when the excitation energy is close to or practically equal to the bound exciton energy. Transitions from the ground and excited states of excitons bound on neutral donor have been found. A model of energy bands explaining the luminescence spectra is proposed. 2. Experimental Investigations were carried out on CuGaS2 samples grown by chemical transport method. The crystals
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represented plates with the area 1 · 1 · 2 cm or prisms with perpendicular faces 8 · 8 · 4 mm. The reflection spectra were measured from surfaces with the c-axis parallel to surface of the crystal. The orientation of the samples and the methods of polarization measurements are described in detail in Ref. [19]. The radiation spectra were measured from the surface (1 1 0). The spectra were measured on a double Raman spectrometer DFS-32 with gratings of 1200 grooves/mm, dispersion 5 A/mm and radiation intensity 1:5. The samples were placed on the cold station of the LTS-22C330 Workhorse-type optical cryogenic system and were kept at the temperature (9.0 ± 0.5) K. The spectra were excited by lines of an Ar+ laser. The spectral position of the lines was determined with the accuracy ±0.2 MeV. 3. Results and discussion In the reflection spectra of CuGaS2 crystals in polarization EIIc, k ? c, the lines n = l (xT = 2.5011 eV, xL = 2.5045 eV), n = 2 (2.5303 eV) and n = 3 (2.5357 eV) of the hydrogen-like series of exciton C4 are found (Fig. 1). Earlier in works [18,19] we have made a technical error in the determination of the Rydberg constant of free excitons. The present paper specifies these data. Exciton binding energy calculated from the energy position of n = 1 and n = 2 lines is equal to 38.5 MeV. Exciton binding energy calculated from the position of lines of excited states (n = 2 and 3) is equal to 39.2 MeV. The Rydberg constant value calculated from lines of the exciton excited states is somewhat higher. Similar values were observed in the exciton spectra of many crystals Cu2O, A2B6. Considering the exciton binding energy being equal to 39.2 MeV, we have determined the band gap energy being equal to 2.5403 eV. From the data on the reflection coefficient measured in the IR range (400 cm1) and near IR range 1 (12,000 cm pffiffi)2the phonon dielectric constant is determined 1þpRffiffi as eb ¼ 1 R . In the region of wavelengths 2.5 lm in
polarization E ? c the dielectric constant is equal to 6. At eb = 72 the reduced effective mass of the C4 excitons equals e R l ¼ RbH ¼ 0; 12m0 . The Bohr radius of the S-state of exci2 tons C4 is equal to aB = 0.3 · 106 cm1, and the longitudinal–transverse splitting equals xLT = 3.9–4.2 MeV. In polarization E ? c, k ? c in the reflection spectra the line n = 1 (xT = 2.5001 eV, xL = 2.5025 eV) of exciton C5 is found [18,19]. In the same polarization (E ? c, k ? c) absorption spectra (Fig. 2) contain the lines n = 1 (2.5001 eV), the lines n = 2 (2.5242 and 2.5252 eV) and the line n = 3 (2.5307 eV) of the C5 exciton. The doublet structure n = 2 is caused by a removal of the orbital degeneration of the highest exciton states C5 under the crystal field action [18,19]. The binding energy of the C5 exciton calculated from the position of lines of the exciton excited states (n = 2 and 3) is equal to 39.4 MeV. It is known that the light I0 incident on a crystal generates in the latter waves determined by the expression [1–8] 1=2 1 n1;2 ðxÞ ¼ e0 þ ½a ða2 þ bÞ1=2 2 where a¼
Mc2 ðx2 x20 þ ixcÞ e1 ; hx0 x2
b¼8
Mc2 e0 xLT hx2
This result corresponds to two branches of polaritons, the upper polariton branch is denoted as xL and the lower one as xT. A phononless ‘‘dead’’ layer of the thickness L exists on the surface. When the light is reflected from two boundaries, the interference effect appears determining the rotation of the contour of the exciton reflection spectrum [23–26]. The formula for the coefficient of light reflection R from the crystal with a ‘‘dead’’ layer has the following form: r12 þ r23 eiu 2 R¼ 1 þ r12 r23 eiu
polarization EIIc the reflection coefficient is equal to R = 0.21, and the dielectric constant is equal to eb = 7. In
where r12 is the constant, r23 is the function having the resonance contour and determining the exciton reflection spectra contour,
Fig. 1. Reflection spectra of CuGaS2 crystals at 10 K in polarization EIIc k ? c.
Fig. 2. Absorption spectra of CuGaS2 crystals at 10 K in polarizations E ? c, k ? c.
N.N. Syrbu et al. / Optical Materials 30 (2007) 451–456
u¼
4pn1 l k
where k is the light wavelength. It is seen from these expressions that R periodically changes with l changing with the period k/2n1. Near the transition x0 the dielectric permeability is: ! xLT eðx; kÞ ¼ eb? 1 þ h2 k2 x þ x0 2M The calculation of the contours of the reflection spectra near the resonance requires to take into account the spatial dispersion (finiteness of the exciton mass M) and boundary conditions on the crystal surface [23–26]. The finiteness of M results in the appearance of normal waves in the region of longitudinal–transverse splitting between the frequencies x0 and xL. The calculations of contours of the reflection spectra in CuGaS2 crystals are carried out considering the theory of additional light waves taking into account the excitonless ‘‘dead’’ layer on the crystal surface [23–26]. For the line n = 1 of the C4 excitons the best agreement of the experimental curve with the calculated one is observed at the following parameters: eb = 7, c = 0.3 MeV, x0 = ˚ and M = 2m0. 2.5006 eV, xLT = 4 MeV, L = 12 A Taking into account the ratios M ¼ mV þ mc and 1 ¼ m1 þ m1c for the effective masses of electrons, the masses l V of light and heavy holes are determined. For the exciton mass M = 2m0 follows mc ¼ 0:13m0 , for M = 2.5m0, mc ¼ 0:122m0 , and for M = 3.5m0 mc ¼ 0:121m0 . It is seen from these data that even at big errors of M determination the mass mc changes weakly. The translational mass M for the C4 excitons was determined from calculation of contours of the reflection spectra with the error ±0.2m0, and it is equal to 2m0. Therefore, at M = 2m0 the value of the electrons mass is equal to mc ¼ ð0:13 0:02Þm0 , and the mass of the holes of the upper valence band is equal to mV 1 ¼ ð1:87 0:02Þm0 [19].
453
Thus, the energy of the transverse C4 exciton is equal to 2.5011 eV, and that of the long-wave free exciton C5 is equal to 2.5001 eV [18,19]. The latter value coincides with the exciton energy 2.5002 eV given in work [12]. The energy of the forbidden C3 exciton according to the data of [12] is equal to 2.4995 eV. Fig. 2 shows a peculiarity from the long-wave side of the transverse exciton C5 at the energy 2,4986 eV. We consider that this is a manifestation of the C3 exciton. In CuGaS2 crystals, the energies xT and xL of the ground states of excitons C5, C4 and C3 are found in very narrow energy interval (1–2 MeV). A correct determination of longitudinal and transverse modes of the ground and excited states of excitons becomes possible by due to the fact that the C4 exciton is allowed in E k c polarization, and the C5 exciton is allowed in E ? c polarization. Besides, the absorption coefficient of the C4 excitons at the frequencies xT is by a factor of 3 Æ 103 higher than the absorption of the C5 excitons [18–21]. Therefore, the determination of parameters of the C4 exciton is carried out from the reflection spectra investigations, and the parameters of the C5 exciton are determined from measurements of absorption spectra in E ? c polarization [18,19]. Fig. 3 shows the radiation spectra of CuGaS2 crystals at 9 K in the geometry X ðZZÞX under the excitation by the ˚ of Ar+ laser. The excitation energy 2.4968 eV line 4965 A is less than the energy of the transverse long-wave excitons C4, C5 and C3. Under these conditions of excitation, as a rule, the Raman scattering instead of the luminescence is observed, since the excitation energy gets into the region of the crystal transparency. In the radiation spectra (Fig. 3) intense narrow line L1 (2.4948 eV), line L2 (2.4938 eV), L3 (2.4933 eV) and L4 (2.4930 eV) are observed. The intense radiation line L1 is 2.1 MeV apart the laser excitation line. The radiation line L4 is 3.8 MeV apart the laser radiation line. Therefore, the whole group of lines L1–L4 cannot be explained by the Raman scatter-
˚ line of Ar+ laser. Fig. 3. Luminescence spectra of CuGaS2 crystals at 10 K and at excitation by 4965 A
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ing. Numerous studies of the Raman scattering in these crystals show that in the spectra the intense mode A1 at 312 cm1 is observed, and other oscillation modes have weak intensity [27,28]. The minimal energy of the optic phonon in these crystals is equal to 8 MeV [27,28]. Hence, the lines found in this work are related to the luminescence spectra. The excitation of the luminescence spectra is possible due to the excitation energy being very close to the optic transition energy. The found radiation spectra are caused by the excitons bound on neutral donor (acceptor). In work [16] the lines 2.495 eV (Ib), 2.493 eV (Ic) are found, being apparently analogues of lines L1 and L2 shown in Fig. 3. The bands C1 (2.4584 eV) and C2 (2.4561 eV) presented in Fig. 3 may be analogues of bands H (2.457 eV), H * (2.460 eV) revealed by the authors of the work [16]. The band Ia (2.501 eV) in the work [16] is attributed to bound excitons. According to the above given data and the results reported in the works [18,19], the band at the energy 2.501 eV is determined by free excitons, namely by the transverse mode of the C4 exciton polariton. The authors of [16,26] attribute short-wave radiation lines (2.495–2.493 eV) to excitons bound on neutral donor (acceptor). The studies of splitting of the lines 2.4939 eV and 2.4920 eV in the magnetic field carried out in work [22] have confirmed this interpretation. In the samples investigated in this work in the region of longer waves from the lines L1–L4, a narrow group of lines g1 (2.4875 eV), g2 (2.4861 eV), g3 (2.4851 eV) and group b1 (2.4655 eV), b2 (2.4643 eV), b3 (2.4636 eV), b4 (2.4631 eV) are observed (Fig. 3). Between these groups of lines wider radiation lines (1–4) are observed, having apparently another nature, i.e. they are not caused by bound excitons of the center under discussion. In the long-wave region from the band b4 narrower radiation lines C1 (2.4584 eV), C2 (2.4561 eV), C3 (2.4542 eV), C4 (2.4528 eV) and wider bands at the energy of 2.4514 eV (C5) and 2.4485 (C6) eV are found. Fig. 4 shows the luminescence spectra of CuGaS2 crys˚ of Ar+ laser. In the region tals excited by the line 4880 A of energies exceeding n = 1 of the free C4 exciton resonance Raman scattering of 1LO, 2LO exciton polaritons is found.
Fig. 4. Luminescence spectra of CuGaS2 crystals at 10 K and at excitation ˚ line of Ar+ laser. by 4880 A
We have discussed these spectra in the work [17], so they are not considered in the present paper. Fig. 4 shows the radiation spectra in the energy range E < En=1 (of free exciton 2.5001 eV). In the radiation spectra intense lines at 2.4948 eV (L1), 2.4938 eV (L2), 2.4933 eV (L3), 2.4930 eV (L4) 2.4875 eV (g1) and 2.486 l eV (g2) are found. Dashed line indicates the position of the line g3, which is found at excitation by the line 496.5 nm and is shown in Fig. 3. In the long-wave region lines being weaker in intensity in the interval 2.4779–2.4318 eV are observed (Fig. 4). The radiation lines L1–L4, g1, g2 shown in Fig. 4 coincide in energy with the energy position of short-wave lines presented in Fig. 3. This confirms the fact that they are caused by luminescence and they are not Raman scattering lines, except for the line g3. The line g3 is 94 cm1 apart the ˚ ). This value coincides frequency of the laser line (4965 A 1 with the frequency (94 cm ) of the symmetry phonon A2 [27–29]. Thus, in CuGaS2 crystals at excitation by lines 4880 and ˚ the luminescence from the bound exciton levels is 4965 A observed. The excitons are bound on neutral donor (acceptor) [16,22]. Analogous spectra are investigated in detail in CdS crystals [6]. In the radiation spectra transitions from the bound excitons levels to the ground and excited states of donor (acceptor) are possible. It is known that the hydrogen-like donor (acceptor) binding energy may be estimated from the ratio 1 G me ¼ 1þ ; ED mh where G is the exciton binding energy, ED is the donor (acceptor) binding energy, me , mh is the effective mass of electrons and holes, respectively [8]. In CdS the ratio mh =me 4. Hopfield [30] has shown that excitons may be bound on donor (acceptor), when the condition mh =me > 1; 4 is realized. For CuGaS2 crystals the effective mass of holes mh is equal to 1.87m0, and the effective mass of electrons me is equal to 0.12m0 [18,19]. The binding energy of the free C4 exciton is equal to 39.2 MeV, while that of the C5 exciton C5 equals 39.4 MeV. Therefore, the ratio of the effective masses of electrons and holes in CuGaS2 crystals is equal to 15.6. Estimating by the above given dependence the binding energy of donor (acceptor) ED (EA) in CuGaS2 crystals, taking into account parameters of excitons C4 we will obtain that it may achieve (41– 42) MeV. Proceeding from these data and the luminescence spectra (Figs. 3, 4) one can suppose that the excitons bound on neutral donor (acceptor) have the energy diagram of the levels shown in Fig. 5. The short-wave radiation lines L1– L4 are determined by transitions from the bound exciton level to the donor (acceptor) ground state level. The second group of narrow lines b1–b4 is caused by the transitions from the bound exciton levels to the donor (acceptor) excited levels. Thus, if the radiation line L1 is caused by the transitions I1c–1S, the line L2 comes from the transitions I1b–1S, the
N.N. Syrbu et al. / Optical Materials 30 (2007) 451–456
455
In this coordinate system the nonmagnetic part of the Hamiltonian simplifies to H¼
~ e2 p02 0 þ Ha 2me m? e0 r
ð4Þ
where H a ¼ ðe2 =e0 Þf½x02 þ y 02 þ z02 ð1 aÞ a ¼ 1 e? m? =eII mII
1=2
r01 g; ð5Þ
Ha is a small perturbation, when the anisotropy factor a is small. For a = 0, the donor energy levels in zero magnetic field are hydrogenic. The donor binding energy for a = 0 is given by E0 ¼ EH m? =e0? e0II Fig. 5. Energy levels and electron transitions of exciton bound on neutral donor (acceptor).
line L3 is related to the transitions I1a–1S and the line L4 is caused by the transitions I10–1S, then the energy intervals I10–I1a, I1a–I1b and I1b–I1c are equal to 1.0, 0.5 and 0.3 MeV, respectively. The radiation maxima b1 (2.4655 eV), b2 (2.4643 eV), b3 (2.4636 eV) and b4 (2.4631 eV) are apparently caused by transitions from the bound exciton levels to the donor (acceptor) excited levels. We consider that the long-wave radiation line (b4) is connected with the transitions from I10 to the level 2Pz. The short-wave radiation line b1 is caused by the transitions I1c to the levels 2S. The radiation line b2 is apart the line b1 by the energy distance of 1.2 MeV. If we consider that b2 is caused by the transitions I1b–2Dx, and the radiation line b3 is caused by the transitions I1a–2Py, we will obtain that the distance between the excited levels of the donor (acceptor) is equal to 0.2 MeV (Fig. 5). In this case the energy distance between the levels 1S–2S is equal to 29.3 MeV. The theory of the energy levels and magnetic splitting of an exciton in a slightly anisotropic crystals was worked out by Hopfield and Thomas [30] and by Wheeler and Dimmock [31]. These theories apply equally well to a donor in CdS. The effective mass Hamiltonian for the donor electron at the zone center in a uniaxial crystal, with the c axis along the z direction, is given by [31] H¼
2 2 ½~ p? ðe=cÞ~ A? ½p ðe=cÞAz þ z 2m? me 2mII me 2 e ~? ~ þ g? bH S ? þ gII bH z S z e0 ðx2 þ y 2 þ z2 e? =eII Þ
ð1Þ
where 1 ~ ~ ~ rÞ; b ¼ e h=2me c; A ¼ ðH 2
e0 ðe? eII Þ
ð2Þ
Wheeler and Dimmock [31] make the coordinate transformation ðx; y; zÞ ¼ ½x0 ; y 0 ; z0 ðm? =mII Þ1=2
ð3Þ
ð6Þ
where EH is the hydrogen atom binding energy. Using first-order perturbation theory, Wheeler and Dimmock calculate the donor energy levels, for a magnetic field in the z direction, to be 1 1 1 E1s ¼ E0 1 þ a þ a2 þ rH 2z gII bH z 3 20 2 1 1 1 2 1 E2s ¼ E0 1 þ a þ a þ 14rH 2z gII bH z 4 3 20 2 1 3 9 2 1 E2pz ¼ E0 1 þ a þ a þ 6r2z gII bH z 4 5 28 2 1 1 9 2 a E2px ¼ E2py ¼ E0 1 þ a þ 4 5 140 1 þ 12rH 2z gII bH z bH =m? ð7Þ 2 where r, the constant determining the diamagnetic shift, is given by r ¼ b2 =2m2? E0
ð8Þ
We can estimate the binding energy of the donor acceptor by adding the average binding energy of the 2p states to our measured average energy separation between the 2p and 1s states. According to Eqs. (6) and (7), the average binding energy of the 2p states is 1 1 3 ED ð2pÞ ¼ ðEH m? =e0II e0? Þ 1 þ a þ a2 ð9Þ 4 3 20 The average energy separation of the 2p and 1s levels is 29.30 ± 0.02 MeV (Fig. 5). To evaluate the ED(2p) energy from the Eq. (9) we can use our measured values of m? and a. We have calculated the value of the electron effective mass m? being equal to 0.13m0, the dielectric constants e0? and e0II equal to 6 and 7, respectively, for polarization E ? c and EIIc. Considering the anisotropy of the electron effective mass m0II m0? being absent, we obtain the anisotropy factor e? m? 0:14 a¼1 eII mII
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Table 1 The energy of the experimentally observed emission lines and their interpretation Free excitons
Bound excitons
Symmetry
Exciton states
Energy, eV, 10 K
Radiation lines
Energy, eV
Transitions
Radiation lines
Energy, eV
C4
n=1 n=2 n=3 n=1
2.5011 2.5303 2.5357 2.5403
L1 L2 L3 L4
2.4948 2.4938 2.4933 2.4930
I1c–1S I1b–1S I1a–1S I10–1S
g1 g2 g3 C1
2.4875 2.4861 2.4851 2.4584
C5
n=1 n=2 n=3 n=1
2.5001 2.5242 2.5307 2.5395
b1 b2 b3 b4
2.4655 2.4643 2.4636 2.4631
I1c–2S I1b–2Px I1a–2Py I10–Pz
C2 C3 C4 C5
2.4561 2.4542 2.4528 2.4514
C6
2.4485
Thus, using the obtained value of a and the above given parameters we have determined the ED(2p) state binding energy being equal to 11.9 MeV and the donor (acceptor) binding energy equal to 41.2 MeV (29.3 + 11.9). One more group of intense lines g1–g3 is observed in the luminescence spectra (Figs. 3, 5). Apparently, these lines are also determined by the excitons bound on donor or acceptor, but this is another center. If this suggestion is correct, these lines take place between the bound exciton levels and the donor ground state. The energy of the experimentally observed emission lines and their interpretation are summarized in the Table 1. Let us note in conclusion that in CuGaS2 crystals intense radiation lines are found, they are interpreted by excitons bound on neutral donor (acceptor). Transitions between the neutral donor (acceptor) ground state and excited states of the bound exciton complex are found in the luminescence spectra. Transitions between the ground and excited states of the neutral donor (acceptor) are also found in the radiation spectra. The main parameters of excitons bound on neutral donor (acceptor) are determined, and the energy distances between the excited states of donor (acceptor) 2S, 2Px, 2Py and 2Pz are estimated. References [1] C.H. Henry, J.J. Hopfield, L.C. Luther, Phys. Rev. Lett. 17 (1966) 1178. [2] G.B. Wright, A. Mooradian, Phys. Rev. Lett. 18 (1967) 608. [3] I.M. Gherlowe, R.L. Aggarwal, B. Lax, Phys. Rev. 7 (1973) 4547. [4] A. Mooradian, G.B. Wright, Phys. Rev. Lett. 18 (1967) 688. [5] P.J. Colwell, M.V. Klein, Phys. Rev. 6 (1972) 498. [6] C.H. Henry, K. Nassau, Phys. Rev. 2 (1970) 997. [7] R.G. Ulbrih, Nguyen van Heu, G. Weisbuch, Phys. Rev. Lett. 46 (1981) 53. [8] J.J. Hopfield, M. Hulin, (Eds.), in: Proceedings of the Seventh International Conference on the Physics of Semiconductors, Paris 725 (1964).
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Resonance impurity radiation in CuGaS2 crystals N.N. Syrbu a, L.L. Nemerenco b
a,*
, V.E. Tezlevan
b
a Technical University of Moldova, MD-2004 Chisinau, Republic of Moldova Institute of Applied Physics, Academy of Sciences of Moldova, MD-2028, Chisinau, Republic of Moldova
Received 3 October 2006; received in revised form 6 December 2006; accepted 7 December 2006 Available online 26 January 2007
Abstract In this work the reflectivity spectra of the CuGaS2 crystals have been investigated in the EIIc polarization and the absorption spectra were studied in the E ? c polarization at the 10 K. The parameters of the excitons and the band gaps have been determined. The low ˚ and 4965 A ˚ temperature (10 K) luminescence spectra of the CuGaS2 crystals have been investigated under the excitation by 4880 A Ar+ laser lines. A model of optic transitions between the levels of excitons bound on neutral donor is proposed. 2007 Elsevier B.V. All rights reserved. Keywords: CuGaS2; Photoluminescence; Absorption; Reflection; Exciton bound on neutral donor (acceptor)
1. Introduction In a number of crystals the resonance luminescence is studied, where laser light is elastically scattered in the crystal leaving donors or acceptors in the excited state [1–8]. The investigation of radiation of bound excitons under the excitation of laser light with the energy close to or equal to the bound exciton (resonance excitation) presents especial interest. The CuGaS2 compound is characterized by anisotropy of optic properties and intensive luminescence on bound excitons. This compound belongs to the chalcopyrite group with space group D12 2d [9–12]. In these crystals resonance Raman scattering of exciton polaritons [13–17], interference of additional exciton waves, spatial dispersion of exciton polaritons [18–20] and biexciton radiation [21] are found. In CuGaS2 crystals three exciton series A, B, C are found [12,17–19]. Interaction of electrons of the conduction band C6 and holes of band C7 is determined by the product of irreducible representations C1 · C6 · C7 = C3 + C4 + C5. As a result of this interaction in the longwave region the C4 exciton allowed in EIIc polarization, the C5 exciton allowed in E ? c polarization and the C3 *
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exciton forbidden in both polarizations are formed. The polariton radiation of the C4 exciton was found and the Raman scattering under the excitation with various laser lines which energy exceeds the longitudinal exciton energy was investigated [9–12]. A series of narrow lines attributed to bound excitons was found in the long-wave region from the transverse exciton frequency [13–16]. However, the bound excitons have not been studied in detail and the structure of zero phonon lines and their replicas have not been revealed. The luminescence excited by the energy less than the band gap (resonance luminescence of bound excitons) has not been reported. We have studied the luminescence of CuGaS2 crystals at excitation energy E < Eg and E being less than the energy of free exciton ground state radiation. We have considered the case when the excitation energy is close to or practically equal to the bound exciton energy. Transitions from the ground and excited states of excitons bound on neutral donor have been found. A model of energy bands explaining the luminescence spectra is proposed. 2. Experimental Investigations were carried out on CuGaS2 samples grown by chemical transport method. The crystals
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represented plates with the area 1 · 1 · 2 cm or prisms with perpendicular faces 8 · 8 · 4 mm. The reflection spectra were measured from surfaces with the c-axis parallel to surface of the crystal. The orientation of the samples and the methods of polarization measurements are described in detail in Ref. [19]. The radiation spectra were measured from the surface (1 1 0). The spectra were measured on a double Raman spectrometer DFS-32 with gratings of 1200 grooves/mm, dispersion 5 A/mm and radiation intensity 1:5. The samples were placed on the cold station of the LTS-22C330 Workhorse-type optical cryogenic system and were kept at the temperature (9.0 ± 0.5) K. The spectra were excited by lines of an Ar+ laser. The spectral position of the lines was determined with the accuracy ±0.2 MeV. 3. Results and discussion In the reflection spectra of CuGaS2 crystals in polarization EIIc, k ? c, the lines n = l (xT = 2.5011 eV, xL = 2.5045 eV), n = 2 (2.5303 eV) and n = 3 (2.5357 eV) of the hydrogen-like series of exciton C4 are found (Fig. 1). Earlier in works [18,19] we have made a technical error in the determination of the Rydberg constant of free excitons. The present paper specifies these data. Exciton binding energy calculated from the energy position of n = 1 and n = 2 lines is equal to 38.5 MeV. Exciton binding energy calculated from the position of lines of excited states (n = 2 and 3) is equal to 39.2 MeV. The Rydberg constant value calculated from lines of the exciton excited states is somewhat higher. Similar values were observed in the exciton spectra of many crystals Cu2O, A2B6. Considering the exciton binding energy being equal to 39.2 MeV, we have determined the band gap energy being equal to 2.5403 eV. From the data on the reflection coefficient measured in the IR range (400 cm1) and near IR range 1 (12,000 cm pffiffi)2the phonon dielectric constant is determined 1þpRffiffi as eb ¼ 1 R . In the region of wavelengths 2.5 lm in
polarization E ? c the dielectric constant is equal to 6. At eb = 72 the reduced effective mass of the C4 excitons equals e R l ¼ RbH ¼ 0; 12m0 . The Bohr radius of the S-state of exci2 tons C4 is equal to aB = 0.3 · 106 cm1, and the longitudinal–transverse splitting equals xLT = 3.9–4.2 MeV. In polarization E ? c, k ? c in the reflection spectra the line n = 1 (xT = 2.5001 eV, xL = 2.5025 eV) of exciton C5 is found [18,19]. In the same polarization (E ? c, k ? c) absorption spectra (Fig. 2) contain the lines n = 1 (2.5001 eV), the lines n = 2 (2.5242 and 2.5252 eV) and the line n = 3 (2.5307 eV) of the C5 exciton. The doublet structure n = 2 is caused by a removal of the orbital degeneration of the highest exciton states C5 under the crystal field action [18,19]. The binding energy of the C5 exciton calculated from the position of lines of the exciton excited states (n = 2 and 3) is equal to 39.4 MeV. It is known that the light I0 incident on a crystal generates in the latter waves determined by the expression [1–8] 1=2 1 n1;2 ðxÞ ¼ e0 þ ½a ða2 þ bÞ1=2 2 where a¼
Mc2 ðx2 x20 þ ixcÞ e1 ; hx0 x2
b¼8
Mc2 e0 xLT hx2
This result corresponds to two branches of polaritons, the upper polariton branch is denoted as xL and the lower one as xT. A phononless ‘‘dead’’ layer of the thickness L exists on the surface. When the light is reflected from two boundaries, the interference effect appears determining the rotation of the contour of the exciton reflection spectrum [23–26]. The formula for the coefficient of light reflection R from the crystal with a ‘‘dead’’ layer has the following form: r12 þ r23 eiu 2 R¼ 1 þ r12 r23 eiu
polarization EIIc the reflection coefficient is equal to R = 0.21, and the dielectric constant is equal to eb = 7. In
where r12 is the constant, r23 is the function having the resonance contour and determining the exciton reflection spectra contour,
Fig. 1. Reflection spectra of CuGaS2 crystals at 10 K in polarization EIIc k ? c.
Fig. 2. Absorption spectra of CuGaS2 crystals at 10 K in polarizations E ? c, k ? c.
N.N. Syrbu et al. / Optical Materials 30 (2007) 451–456
u¼
4pn1 l k
where k is the light wavelength. It is seen from these expressions that R periodically changes with l changing with the period k/2n1. Near the transition x0 the dielectric permeability is: ! xLT eðx; kÞ ¼ eb? 1 þ h2 k2 x þ x0 2M The calculation of the contours of the reflection spectra near the resonance requires to take into account the spatial dispersion (finiteness of the exciton mass M) and boundary conditions on the crystal surface [23–26]. The finiteness of M results in the appearance of normal waves in the region of longitudinal–transverse splitting between the frequencies x0 and xL. The calculations of contours of the reflection spectra in CuGaS2 crystals are carried out considering the theory of additional light waves taking into account the excitonless ‘‘dead’’ layer on the crystal surface [23–26]. For the line n = 1 of the C4 excitons the best agreement of the experimental curve with the calculated one is observed at the following parameters: eb = 7, c = 0.3 MeV, x0 = ˚ and M = 2m0. 2.5006 eV, xLT = 4 MeV, L = 12 A Taking into account the ratios M ¼ mV þ mc and 1 ¼ m1 þ m1c for the effective masses of electrons, the masses l V of light and heavy holes are determined. For the exciton mass M = 2m0 follows mc ¼ 0:13m0 , for M = 2.5m0, mc ¼ 0:122m0 , and for M = 3.5m0 mc ¼ 0:121m0 . It is seen from these data that even at big errors of M determination the mass mc changes weakly. The translational mass M for the C4 excitons was determined from calculation of contours of the reflection spectra with the error ±0.2m0, and it is equal to 2m0. Therefore, at M = 2m0 the value of the electrons mass is equal to mc ¼ ð0:13 0:02Þm0 , and the mass of the holes of the upper valence band is equal to mV 1 ¼ ð1:87 0:02Þm0 [19].
453
Thus, the energy of the transverse C4 exciton is equal to 2.5011 eV, and that of the long-wave free exciton C5 is equal to 2.5001 eV [18,19]. The latter value coincides with the exciton energy 2.5002 eV given in work [12]. The energy of the forbidden C3 exciton according to the data of [12] is equal to 2.4995 eV. Fig. 2 shows a peculiarity from the long-wave side of the transverse exciton C5 at the energy 2,4986 eV. We consider that this is a manifestation of the C3 exciton. In CuGaS2 crystals, the energies xT and xL of the ground states of excitons C5, C4 and C3 are found in very narrow energy interval (1–2 MeV). A correct determination of longitudinal and transverse modes of the ground and excited states of excitons becomes possible by due to the fact that the C4 exciton is allowed in E k c polarization, and the C5 exciton is allowed in E ? c polarization. Besides, the absorption coefficient of the C4 excitons at the frequencies xT is by a factor of 3 Æ 103 higher than the absorption of the C5 excitons [18–21]. Therefore, the determination of parameters of the C4 exciton is carried out from the reflection spectra investigations, and the parameters of the C5 exciton are determined from measurements of absorption spectra in E ? c polarization [18,19]. Fig. 3 shows the radiation spectra of CuGaS2 crystals at 9 K in the geometry X ðZZÞX under the excitation by the ˚ of Ar+ laser. The excitation energy 2.4968 eV line 4965 A is less than the energy of the transverse long-wave excitons C4, C5 and C3. Under these conditions of excitation, as a rule, the Raman scattering instead of the luminescence is observed, since the excitation energy gets into the region of the crystal transparency. In the radiation spectra (Fig. 3) intense narrow line L1 (2.4948 eV), line L2 (2.4938 eV), L3 (2.4933 eV) and L4 (2.4930 eV) are observed. The intense radiation line L1 is 2.1 MeV apart the laser excitation line. The radiation line L4 is 3.8 MeV apart the laser radiation line. Therefore, the whole group of lines L1–L4 cannot be explained by the Raman scatter-
˚ line of Ar+ laser. Fig. 3. Luminescence spectra of CuGaS2 crystals at 10 K and at excitation by 4965 A
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ing. Numerous studies of the Raman scattering in these crystals show that in the spectra the intense mode A1 at 312 cm1 is observed, and other oscillation modes have weak intensity [27,28]. The minimal energy of the optic phonon in these crystals is equal to 8 MeV [27,28]. Hence, the lines found in this work are related to the luminescence spectra. The excitation of the luminescence spectra is possible due to the excitation energy being very close to the optic transition energy. The found radiation spectra are caused by the excitons bound on neutral donor (acceptor). In work [16] the lines 2.495 eV (Ib), 2.493 eV (Ic) are found, being apparently analogues of lines L1 and L2 shown in Fig. 3. The bands C1 (2.4584 eV) and C2 (2.4561 eV) presented in Fig. 3 may be analogues of bands H (2.457 eV), H * (2.460 eV) revealed by the authors of the work [16]. The band Ia (2.501 eV) in the work [16] is attributed to bound excitons. According to the above given data and the results reported in the works [18,19], the band at the energy 2.501 eV is determined by free excitons, namely by the transverse mode of the C4 exciton polariton. The authors of [16,26] attribute short-wave radiation lines (2.495–2.493 eV) to excitons bound on neutral donor (acceptor). The studies of splitting of the lines 2.4939 eV and 2.4920 eV in the magnetic field carried out in work [22] have confirmed this interpretation. In the samples investigated in this work in the region of longer waves from the lines L1–L4, a narrow group of lines g1 (2.4875 eV), g2 (2.4861 eV), g3 (2.4851 eV) and group b1 (2.4655 eV), b2 (2.4643 eV), b3 (2.4636 eV), b4 (2.4631 eV) are observed (Fig. 3). Between these groups of lines wider radiation lines (1–4) are observed, having apparently another nature, i.e. they are not caused by bound excitons of the center under discussion. In the long-wave region from the band b4 narrower radiation lines C1 (2.4584 eV), C2 (2.4561 eV), C3 (2.4542 eV), C4 (2.4528 eV) and wider bands at the energy of 2.4514 eV (C5) and 2.4485 (C6) eV are found. Fig. 4 shows the luminescence spectra of CuGaS2 crys˚ of Ar+ laser. In the region tals excited by the line 4880 A of energies exceeding n = 1 of the free C4 exciton resonance Raman scattering of 1LO, 2LO exciton polaritons is found.
Fig. 4. Luminescence spectra of CuGaS2 crystals at 10 K and at excitation ˚ line of Ar+ laser. by 4880 A
We have discussed these spectra in the work [17], so they are not considered in the present paper. Fig. 4 shows the radiation spectra in the energy range E < En=1 (of free exciton 2.5001 eV). In the radiation spectra intense lines at 2.4948 eV (L1), 2.4938 eV (L2), 2.4933 eV (L3), 2.4930 eV (L4) 2.4875 eV (g1) and 2.486 l eV (g2) are found. Dashed line indicates the position of the line g3, which is found at excitation by the line 496.5 nm and is shown in Fig. 3. In the long-wave region lines being weaker in intensity in the interval 2.4779–2.4318 eV are observed (Fig. 4). The radiation lines L1–L4, g1, g2 shown in Fig. 4 coincide in energy with the energy position of short-wave lines presented in Fig. 3. This confirms the fact that they are caused by luminescence and they are not Raman scattering lines, except for the line g3. The line g3 is 94 cm1 apart the ˚ ). This value coincides frequency of the laser line (4965 A 1 with the frequency (94 cm ) of the symmetry phonon A2 [27–29]. Thus, in CuGaS2 crystals at excitation by lines 4880 and ˚ the luminescence from the bound exciton levels is 4965 A observed. The excitons are bound on neutral donor (acceptor) [16,22]. Analogous spectra are investigated in detail in CdS crystals [6]. In the radiation spectra transitions from the bound excitons levels to the ground and excited states of donor (acceptor) are possible. It is known that the hydrogen-like donor (acceptor) binding energy may be estimated from the ratio 1 G me ¼ 1þ ; ED mh where G is the exciton binding energy, ED is the donor (acceptor) binding energy, me , mh is the effective mass of electrons and holes, respectively [8]. In CdS the ratio mh =me 4. Hopfield [30] has shown that excitons may be bound on donor (acceptor), when the condition mh =me > 1; 4 is realized. For CuGaS2 crystals the effective mass of holes mh is equal to 1.87m0, and the effective mass of electrons me is equal to 0.12m0 [18,19]. The binding energy of the free C4 exciton is equal to 39.2 MeV, while that of the C5 exciton C5 equals 39.4 MeV. Therefore, the ratio of the effective masses of electrons and holes in CuGaS2 crystals is equal to 15.6. Estimating by the above given dependence the binding energy of donor (acceptor) ED (EA) in CuGaS2 crystals, taking into account parameters of excitons C4 we will obtain that it may achieve (41– 42) MeV. Proceeding from these data and the luminescence spectra (Figs. 3, 4) one can suppose that the excitons bound on neutral donor (acceptor) have the energy diagram of the levels shown in Fig. 5. The short-wave radiation lines L1– L4 are determined by transitions from the bound exciton level to the donor (acceptor) ground state level. The second group of narrow lines b1–b4 is caused by the transitions from the bound exciton levels to the donor (acceptor) excited levels. Thus, if the radiation line L1 is caused by the transitions I1c–1S, the line L2 comes from the transitions I1b–1S, the
N.N. Syrbu et al. / Optical Materials 30 (2007) 451–456
455
In this coordinate system the nonmagnetic part of the Hamiltonian simplifies to H¼
~ e2 p02 0 þ Ha 2me m? e0 r
ð4Þ
where H a ¼ ðe2 =e0 Þf½x02 þ y 02 þ z02 ð1 aÞ a ¼ 1 e? m? =eII mII
1=2
r01 g; ð5Þ
Ha is a small perturbation, when the anisotropy factor a is small. For a = 0, the donor energy levels in zero magnetic field are hydrogenic. The donor binding energy for a = 0 is given by E0 ¼ EH m? =e0? e0II Fig. 5. Energy levels and electron transitions of exciton bound on neutral donor (acceptor).
line L3 is related to the transitions I1a–1S and the line L4 is caused by the transitions I10–1S, then the energy intervals I10–I1a, I1a–I1b and I1b–I1c are equal to 1.0, 0.5 and 0.3 MeV, respectively. The radiation maxima b1 (2.4655 eV), b2 (2.4643 eV), b3 (2.4636 eV) and b4 (2.4631 eV) are apparently caused by transitions from the bound exciton levels to the donor (acceptor) excited levels. We consider that the long-wave radiation line (b4) is connected with the transitions from I10 to the level 2Pz. The short-wave radiation line b1 is caused by the transitions I1c to the levels 2S. The radiation line b2 is apart the line b1 by the energy distance of 1.2 MeV. If we consider that b2 is caused by the transitions I1b–2Dx, and the radiation line b3 is caused by the transitions I1a–2Py, we will obtain that the distance between the excited levels of the donor (acceptor) is equal to 0.2 MeV (Fig. 5). In this case the energy distance between the levels 1S–2S is equal to 29.3 MeV. The theory of the energy levels and magnetic splitting of an exciton in a slightly anisotropic crystals was worked out by Hopfield and Thomas [30] and by Wheeler and Dimmock [31]. These theories apply equally well to a donor in CdS. The effective mass Hamiltonian for the donor electron at the zone center in a uniaxial crystal, with the c axis along the z direction, is given by [31] H¼
2 2 ½~ p? ðe=cÞ~ A? ½p ðe=cÞAz þ z 2m? me 2mII me 2 e ~? ~ þ g? bH S ? þ gII bH z S z e0 ðx2 þ y 2 þ z2 e? =eII Þ
ð1Þ
where 1 ~ ~ ~ rÞ; b ¼ e h=2me c; A ¼ ðH 2
e0 ðe? eII Þ
ð2Þ
Wheeler and Dimmock [31] make the coordinate transformation ðx; y; zÞ ¼ ½x0 ; y 0 ; z0 ðm? =mII Þ1=2
ð3Þ
ð6Þ
where EH is the hydrogen atom binding energy. Using first-order perturbation theory, Wheeler and Dimmock calculate the donor energy levels, for a magnetic field in the z direction, to be 1 1 1 E1s ¼ E0 1 þ a þ a2 þ rH 2z gII bH z 3 20 2 1 1 1 2 1 E2s ¼ E0 1 þ a þ a þ 14rH 2z gII bH z 4 3 20 2 1 3 9 2 1 E2pz ¼ E0 1 þ a þ a þ 6r2z gII bH z 4 5 28 2 1 1 9 2 a E2px ¼ E2py ¼ E0 1 þ a þ 4 5 140 1 þ 12rH 2z gII bH z bH =m? ð7Þ 2 where r, the constant determining the diamagnetic shift, is given by r ¼ b2 =2m2? E0
ð8Þ
We can estimate the binding energy of the donor acceptor by adding the average binding energy of the 2p states to our measured average energy separation between the 2p and 1s states. According to Eqs. (6) and (7), the average binding energy of the 2p states is 1 1 3 ED ð2pÞ ¼ ðEH m? =e0II e0? Þ 1 þ a þ a2 ð9Þ 4 3 20 The average energy separation of the 2p and 1s levels is 29.30 ± 0.02 MeV (Fig. 5). To evaluate the ED(2p) energy from the Eq. (9) we can use our measured values of m? and a. We have calculated the value of the electron effective mass m? being equal to 0.13m0, the dielectric constants e0? and e0II equal to 6 and 7, respectively, for polarization E ? c and EIIc. Considering the anisotropy of the electron effective mass m0II m0? being absent, we obtain the anisotropy factor e? m? 0:14 a¼1 eII mII
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Table 1 The energy of the experimentally observed emission lines and their interpretation Free excitons
Bound excitons
Symmetry
Exciton states
Energy, eV, 10 K
Radiation lines
Energy, eV
Transitions
Radiation lines
Energy, eV
C4
n=1 n=2 n=3 n=1
2.5011 2.5303 2.5357 2.5403
L1 L2 L3 L4
2.4948 2.4938 2.4933 2.4930
I1c–1S I1b–1S I1a–1S I10–1S
g1 g2 g3 C1
2.4875 2.4861 2.4851 2.4584
C5
n=1 n=2 n=3 n=1
2.5001 2.5242 2.5307 2.5395
b1 b2 b3 b4
2.4655 2.4643 2.4636 2.4631
I1c–2S I1b–2Px I1a–2Py I10–Pz
C2 C3 C4 C5
2.4561 2.4542 2.4528 2.4514
C6
2.4485
Thus, using the obtained value of a and the above given parameters we have determined the ED(2p) state binding energy being equal to 11.9 MeV and the donor (acceptor) binding energy equal to 41.2 MeV (29.3 + 11.9). One more group of intense lines g1–g3 is observed in the luminescence spectra (Figs. 3, 5). Apparently, these lines are also determined by the excitons bound on donor or acceptor, but this is another center. If this suggestion is correct, these lines take place between the bound exciton levels and the donor ground state. The energy of the experimentally observed emission lines and their interpretation are summarized in the Table 1. Let us note in conclusion that in CuGaS2 crystals intense radiation lines are found, they are interpreted by excitons bound on neutral donor (acceptor). Transitions between the neutral donor (acceptor) ground state and excited states of the bound exciton complex are found in the luminescence spectra. Transitions between the ground and excited states of the neutral donor (acceptor) are also found in the radiation spectra. The main parameters of excitons bound on neutral donor (acceptor) are determined, and the energy distances between the excited states of donor (acceptor) 2S, 2Px, 2Py and 2Pz are estimated. References [1] C.H. Henry, J.J. Hopfield, L.C. Luther, Phys. Rev. Lett. 17 (1966) 1178. [2] G.B. Wright, A. Mooradian, Phys. Rev. Lett. 18 (1967) 608. [3] I.M. Gherlowe, R.L. Aggarwal, B. Lax, Phys. Rev. 7 (1973) 4547. [4] A. Mooradian, G.B. Wright, Phys. Rev. Lett. 18 (1967) 688. [5] P.J. Colwell, M.V. Klein, Phys. Rev. 6 (1972) 498. [6] C.H. Henry, K. Nassau, Phys. Rev. 2 (1970) 997. [7] R.G. Ulbrih, Nguyen van Heu, G. Weisbuch, Phys. Rev. Lett. 46 (1981) 53. [8] J.J. Hopfield, M. Hulin, (Eds.), in: Proceedings of the Seventh International Conference on the Physics of Semiconductors, Paris 725 (1964).
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